Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group

TL;DR

This paper explores the equivalence between Sion's minimax theorem and Nash equilibrium in a symmetric, zero-sum, multi-player game with two groups, providing conditions for their coexistence and illustrating with an oligopoly example.

Contribution

It establishes the equivalence between Sion's minimax theorem and Nash equilibrium in symmetric, multi-group zero-sum games, linking strategic strategies and equilibrium existence.

Findings

Symmetric Nash equilibrium implies Sion's minimax theorem.

Sion's minimax theorem implies the existence of symmetric Nash equilibrium.

The results are exemplified in an oligopoly profit maximization game.

Abstract

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results. 1. The existence of Nash equilibrium which is symmetric in each group implies Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group. %given the values of the strategic variables. 2. Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of a Nash equilibrium which is symmetric in each group. Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.